Clausius inequality temperature

[LKmPV]

in the clausius inequality is the temperature that of the system or of the surroundings? or is it temperature of the body receiving positive heat?
(assuming the irreversibility is due to heat transfer with finite temperature difference)
[borgnakke and sonntag-principle of entropy increase for control volume]

 

[Chestermiller]

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In the Clausius inequality, T is the temperature at the interface between the system and the surroundings, at the location where dQ is entering. In an irreversible process, the temperature within the system is not usually uniform.

A more precise representation of the Clausius inequality is given by:
$$ΔS≥\int_0^t\int_A{(\frac{\vec{q }}{T})\centerdot \vec{n} dA}dt$$
where $\vec{q}$ is the local heat flux vector at the interface, $\vec{n}$ is an inwardly directed unit normal across the interface drawn from the surroundings to the system, and T is the temperature at the interface. The area integral is over the entire instantaneous interface between the system and the surroundings (the interface may be moving during the process), and the time integral is over the (irreversible) process from the initial thermodynamic equilibrium state to the final equilibrium state.